Statics CEE 331 July 14, 2015 CEE 331 July 14, 2015

StaticsStaticsCEE 331

April 19, 2023

CEE 331

April 19, 2023

Definitions and ApplicationsDefinitions and Applications

Statics: no relative motion between adjacent fluid layers. Shear stress is zero Only _______ can be acting on fluid surfaces

Gravity force acts on the fluid (____ force) Applications:

Pressure variation within a reservoir Forces on submerged surfaces Tensile stress on pipe walls Buoyant forces

Statics: no relative motion between adjacent fluid layers. Shear stress is zero Only _______ can be acting on fluid surfaces

Gravity force acts on the fluid (____ force) Applications:

Pressure variation within a reservoir Forces on submerged surfaces Tensile stress on pipe walls Buoyant forces

pressure

body

Motivation?Motivation?

What are the pressure forces behind the Hoover Dam?

What are the pressure forces behind the Hoover Dam?

Upstream face of Hoover DamUpstream face of Hoover DamUpstream face of Hoover DamUpstream face of Hoover Dam

Upstream face of Hoover Dam in 1935Upstream face of Hoover Dam in 1935

Crest thickness: 13.7 m Base thickness: 201 mWHY???

What do you think?What do you think?

Lake Mead, the lake behind Hoover Dam, is the world's largest artificial body of water by volume (35 km3). Is the pressure at the base of Hoover Dam affected by the volume of water in Lake Mead?

What do we need to know?What do we need to know?

Pressure variation with direction Pressure variation with location How can we calculate the total force on a

submerged surface?

Pressure variation with direction Pressure variation with location How can we calculate the total force on a

submerged surface?

Pressure Variation with Direction(Pascal’s law)

Pressure Variation with Direction(Pascal’s law)

yy

xx

pss

pxy

pyx

y

x

s

2yx

2yx

Body forces

Surface forcesEquation of Motion

xF xF

F = ma

02

m xx ayx

a 02

m xx ayx

a

y sin s y sin s

0y p -y p sx 0y p -y p sx

pxy - pss sin

Independent of direction!

Pressure FieldPressure Field

In the absence of shearing forces (no relative motion between fluid particles) what causes pressure variation within a fluid?

Consider a soda can in space… Throw the soda can to another astronaut… Throw the soda can toward the moon

1 minute… What causes pressure gradients?

In the absence of shearing forces (no relative motion between fluid particles) what causes pressure variation within a fluid?

Consider a soda can in space… Throw the soda can to another astronaut… Throw the soda can toward the moon

1 minute… What causes pressure gradients?

ppz

zx y

FH IK 2

Pressure FieldPressure Field

ppy

yx z

FHG

IKJ

2

m x y zji

zz

yy

xx

k

ppz

zx y

FH IK 2

ppy

yx z

FHG

IKJ

2

Small element of fluid in Small element of fluid in pressure pressure gradientgradient with arbitrary __________. with arbitrary __________.

Forces acting on surfaces of elementPressure is Pressure is pp at at

center of elementcenter of element

acceleration

Mass…Mass…

Same in x!Same in x!Now let’s sum the forces in the y direction

Simplify the expression for the force acting on the element

Simplify the expression for the force acting on the element

Fpy

x y zy

Fpx

x y zx

Fpz

x y zz

F i j k

FHG

IKJ

px

py

pz

x y z

px

py

pz

p i j k

F p x y z

Same in xyz!Same in xyz!

This begs for vector notation!This begs for vector notation!

Forces acting on element of Forces acting on element of fluid due to pressure gradientfluid due to pressure gradient

2 2y

p y p yF p x z p x z

y yd d

d d d d d¶ ¶æ ö æ ö

= - - +ç ÷ ç ÷è ø è ø¶ ¶

Apply Newton’s Second LawApply Newton’s Second Law

F a m

p x y z x y z a

p a

F p x y z

m x y za ad rd d d= Mass of element of fluidMass of element of fluid

Substitute into Newton’s 2nd Law

Obtain a general vector expression Obtain a general vector expression relating pressure gradient to relating pressure gradient to accelerationacceleration and write the 3 component equations.and write the 3 component equations.

Note that we are effectively Note that we are effectively accelerating upward at g when we accelerating upward at g when we are “at rest” on earth’s surface!are “at rest” on earth’s surface!

x y z

p p pa a a

x y zr r r

¶ ¶ ¶= =- -

¶=-

¶ ¶

dpdz

g

ˆp r g- Ñ = +a k Text version of eq.Text version of eq.

At restAt rest

3 component equations3 component equations

Compressible fluid - changing density

Changing gravity

Pressure Variation When the Specific Weight is ConstantPressure Variation When the Specific Weight is Constant

What are the two things that could make specific weight () vary in a fluid?

What are the two things that could make specific weight () vary in a fluid?

= g

Piezometric head

dp dz

dp dzp

p

z

z

1

2

1

2z z

p p z z2 1 2 1 a f pz

pz1

12

2

Constant specific weight!

Generalize to any a!

p hgD =

Example: Pressure at the bottom of a Tank of Water?

Example: Pressure at the bottom of a Tank of Water?

Does the pressure at the bottom of the tank increase if the diameter of the tank increases?

h

11

22

p p z z2 1 2 1 a f

What is the pressure at the top of the tank?What is the pressure at the top of the tank?

Suppose I define pressure and elevation as zero at the water surface. Suppose I define pressure and elevation as zero at the water surface. What is the piezometric head everywhere in the tank? ______What is the piezometric head everywhere in the tank? ______Zero!

No!

6894.76 Pa/psi

Units and Scales of Pressure Measurement

Units and Scales of Pressure Measurement

Standard atmospheric pressure

Local atmospheric pressure

Absolute zero (complete vacuum)

Absolute pressureGage pressure

1 atmosphere101.325 kPa14.7 psi______ m H20760 mm Hg

Suction vacuum(gage pressure)Local baromet

er reading10.3410.34

atmph

g= =

3

3

101 109806 /

x PaN m

Mercury Barometer (team work)Mercury Barometer (team work)

22

11 z

p z

p 2

21

1 zp

zp

6.13HgS 6.13HgSWhat is the local atmospheric pressure (in kPa) when R is 750 mm Hg?

R

1

2

1221 zzp p Hg 1221 zzp p Hg P2 = Hg vapor pressure

R p Hg1 R p Hg1

waterHgHg S waterHgHg S

RS p Hg1 RS p Hg1

PammN p 000,10075.0/98066.13 31 PammN p 000,10075.0/98066.13 31

fluid

water

r

r= fluid

water

r

r=

Assume constant

1000 kg/m3

1 x10-3 N·s/m2

9800 N/m3

101.3 kPa

A few important constants!A few important constants!

Properties of water Density: _______ Viscosity: ___________ Specific weight: _______

Properties of the atmosphere Atmospheric pressure ______ Height of a column of water that can be

supported by atmospheric pressure _____

Properties of water Density: _______ Viscosity: ___________ Specific weight: _______

Properties of the atmosphere Atmospheric pressure ______ Height of a column of water that can be

supported by atmospheric pressure _____10.3 m

Pressure Variation in a Compressible Fluid

Pressure Variation in a Compressible Fluid

Perfect gas at constant temperature (Isothermal)

Perfect gas with constant temperature gradient

Perfect gas at constant temperature (Isothermal)

Perfect gas with constant temperature gradient

Perfect Gas at Constant Temperature (Isothermal)Perfect Gas at Constant

Temperature (Isothermal)

nRTpV nRTpV gaspM

RTgaspM

RT

g g

dzRT

gpMdp gas dz

RT

gpMdp gas

dzRT

gM

pdp

z

z

gasp

p

2

1

2

1

dzRT

gM

pdp

z

z

gasp

p

2

1

2

1

121

2ln zzRT

gM

pp gas 12

1

2ln zzRT

gM

pp gas

e

zzRT

gM gas

pp

12

12

e

zzRT

gM gas

pp

12

12

Mgas is molecular mass

is function of pdp dz

gasnM

Vr = =gasnM

Vr = =

Integrate…Integrate…

= 0.00650 K/m

Perfect Gas with Constant Temperature Gradient

Perfect Gas with Constant Temperature Gradient

The atmosphere can be modeled as having a constant temperature gradient

The atmosphere can be modeled as having a constant temperature gradient

zTT a zTT a

dzRT

gM

pdp gas dz

RT

gM

pdp gas

dzzTR

gM

pdp

a

gas

dzzTR

gM

pdp

a

gas

z

a

gasp

p zTdz

R

gM

pdp

a 0

z

a

gasp

p zTdz

R

gM

pdp

a 0

a

agas

a TzT

R

gM

pp

ln

1ln

a

agas

a TzT

R

gM

pp

ln

1ln

R

gM

aa

gas

Tz

pp

1

R

gM

aa

gas

Tz

pp

1

0

20

40

60

80

100

0 5000 10000 15000Elevation (m)

Pre

ssur

e (k

Pa)

Lapse rateLapse rate

Mt. Everest

Pressure MeasurementPressure Measurement

Barometers Manometers

Standard Differential

Pressure Transducers

Barometers Manometers

Standard Differential

Pressure Transducers

Measure atmospheric pressure

Pressure relative to atm.

Pressure difference between 2 pts.

A

Standard ManometersStandard Manometers

What is the pressure at A given h?

Pressure in water distribution systems commonly varies between 25 and 100 psi (175 to 700 kPa). How high would the water rise in a manometer connected to a pipe containing water at 500 kPa?

What is the pressure at A given h?

Pressure in water distribution systems commonly varies between 25 and 100 psi (175 to 700 kPa). How high would the water rise in a manometer connected to a pipe containing water at 500 kPa?

h

p = h

h = p/h = 500,000 Pa/9800 N/m3

h = 51 m Why is this a reasonable pressure?Why is this a reasonable pressure?

gage

P1 = 0P1 = 0 hh11

??

hh22

Manometers for High PressuresManometers for High Pressures

Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2.

What do you know? _____

Use statics to find other pressures.

Find the gage pressure in the center of the sphere. The sphere contains fluid with 1 and the manometer contains fluid with 2.

What do you know? _____

Use statics to find other pressures.

11

22

33

=P3=P3

1

2

For small h1 use fluid with high density. Mercury!Mercury!

+ h12+ h12 - h21- h21P1P1

- h2Hg- h2Hg- h3w- h3w

Differential ManometersDifferential Manometers

h1

h3

Mercury

Find the drop in pressure between point 1 and point 2.

p1p2Water

h2

orificeorifice

= p2= p2

p1 - p2 = (h3-h1)w + h2Hg

p1 - p2 = h2(Hg - w)

p1p1 + h1w+ h1w

Procedure to keep track of pressures

Procedure to keep track of pressures

Start at a known point or at one end of the system and write the pressure there using an appropriate symbol

Add to this the change in pressure to the next meniscus (plus if the next meniscus is lower, and minus if higher)

Continue until the other end of the gage is reached and equate the expression to the pressure at that point

Start at a known point or at one end of the system and write the pressure there using an appropriate symbol

Add to this the change in pressure to the next meniscus (plus if the next meniscus is lower, and minus if higher)

Continue until the other end of the gage is reached and equate the expression to the pressure at that point

p1 + p = p2

Pressure TransducersPressure Transducers

Excitation: 10 Vdc regulated Output: 100 millivolts Accuracy: ±1% FS Proof Pressure: 140 kPa (20 psi) for 7 kPa model No Mercury! Can be monitored easily by computer Myriad of applications

Volume of liquid in a tank Flow rates Process monitoring and control

Excitation: 10 Vdc regulated Output: 100 millivolts Accuracy: ±1% FS Proof Pressure: 140 kPa (20 psi) for 7 kPa model No Mercury! Can be monitored easily by computer Myriad of applications

Volume of liquid in a tank Flow rates Process monitoring and control

Full Scale

Strain gageStrain gage

What happens to the resistance thru the strain gage if it is stretched in the y direction? ________________

In the x direction? ________________

Strain gage can be made of wire that is then bonded to the objected that is undergoing strain

Or diffused into a crystalline silicon diaphragm (___________)

What happens to the resistance thru the strain gage if it is stretched in the y direction? ________________

In the x direction? ________________

Strain gage can be made of wire that is then bonded to the objected that is undergoing strain

Or diffused into a crystalline silicon diaphragm (___________)

x

y

Little change

Great change

Piezoresistive

Types of Diaphragms Used for Pressure Measurements

Types of Diaphragms Used for Pressure Measurements

Stainless Steel Strain gages bonded to the stainless steel Typical full scale output of 3 mV/V

Piezoresistive Strain gage diffused into silicon wafers Typical full scale output of 10 mV/V

Stainless Steel Strain gages bonded to the stainless steel Typical full scale output of 3 mV/V

Piezoresistive Strain gage diffused into silicon wafers Typical full scale output of 10 mV/V

Piezoresistive DiaphragmsPiezoresistive Diaphragms

Excitation +

Excitation -Signal -

Signal +

R is function of ____________ on crystal and strain.orientation

R+R R+R

R-R

R-R

SiliconSilicon

Ideal material for receiving the applied force

Perfect crystal Returns to its initial shape (no hysteresis) Good elasticity No need for special bonding between

material receiving force and strain gage

Ideal material for receiving the applied force

Perfect crystal Returns to its initial shape (no hysteresis) Good elasticity No need for special bonding between

material receiving force and strain gage

Pressure Sensor FailurePressure Sensor Failure

High pressures – rupture crystal (beware of resulting leak!)

Water hammer – High speed pressure waves (speed of sound) Result from flow transients such as rapidly

shutting valves Install pressure snubber!

Incompatible materials

High pressures – rupture crystal (beware of resulting leak!)

Water hammer – High speed pressure waves (speed of sound) Result from flow transients such as rapidly

shutting valves Install pressure snubber!

Incompatible materials

Absolute vs. Gage vs. Differential

Absolute vs. Gage vs. Differential

Absolute Port 2 sealed with vacuum

on bottom side of silicon crystal

Gage Port 2 open to atmosphere

Differential Both ports connected to

system

Absolute Port 2 sealed with vacuum

on bottom side of silicon crystal

Gage Port 2 open to atmosphere

Differential Both ports connected to

system

Port 1

Port 2

Pressure is independent of Pressure increases with

constant density gas at constant temperature gas with constant temperature gradient

Pressure scales units datum

Pressure measurement

Pressure is independent of Pressure increases with

constant density gas at constant temperature gas with constant temperature gradient

Pressure scales units datum

Pressure measurement

direction

depthp = h

Use ideal gas law

Summary for StaticsSummary for Statics

dzdp dzdp

ReviewReview

Pressure increases or decreases as we move in direction of acceleration vector?

The free surface is _______ to the acceleration vector.

What is an equation that describes the change in pressure with depth in a fluid?

Suppose a tank of fuel is accelerating upward at 2g. What is the change in pressure with depth in the fuel?

Pressure increases or decreases as we move in direction of acceleration vector?

The free surface is _______ to the acceleration vector.

What is an equation that describes the change in pressure with depth in a fluid?

Suppose a tank of fuel is accelerating upward at 2g. What is the change in pressure with depth in the fuel?

normal

dp adzr=-

Statics exampleStatics example

What is the air pressure in the cave air pocket?What is the air pressure in the cave air pocket?

Statics LabStatics Lab

How did the bubbler work? How did the bubbler work?

“Somebody finally got smart and came up with an above-ground pool that’s got a deep end and a shallow end.”

“Somebody finally got smart and came up with an above-ground pool that’s got a deep end and a shallow end.”